Sets with dependent elements: A formalization of Castoriadis' notion of magma

Abstract

We present a formalization of collections that Cornelius Castoriadis calls ``magmas'', especially the property which mainly characterizes them and distinguishes them from the usual cantorian sets. It is the property of their elements to {\em depend} on other elements, either in a one-way or a two-way manner, so that one cannot occur in a collection without the occurrence of those dependent on it. Such a dependence relation can be represented by a pre-order relation $\preccurlyeq$ Then, working in a mild strengthening of the theory ${\rm ZFA}$, where $A$ is an infinite set of atoms equipped with a primitive pre-ordering $\preccurlyeq$, the class of magmas over $A$ is represented by the class $LO(A,\preccurlyeq)$ of nonempty open subsets of $A$ with respect to the lower topology of $\langle A,\preccurlyeq\rangle$. Next the pre-ordering $\preccurlyeq$ is shifted (by a kind of simulation) to a pre-ordering $\preccurlyeq^+$ on ${\cal P}(A)$, which turns out to satisfy the same non-minimality condition as well, and which, happily, when restricted to $LO(A,\preccurlyeq)$ coincides with $\subseteq$. This allows us to define a hierarchy $M_\alpha(A)$, along all ordinals $\alpha\geq 1$, the``magmatic hierarchy'', such that $M_1(A)=LO(A,\preccurlyeq)$, $M_{\alpha+1}(A)=LO(M_\alpha(A),\subseteq)$, and $M_\alpha(A)=\bigcup_{\beta<\alpha}M_\beta(A)$, for a limit ordinal $\alpha$. For every $\alpha\geq 1$, $M_\alpha(A)\subseteq V_\alpha(A)$, where $V_\alpha(A)$ are the levels of the universe $V(A)$ of ${\rm ZFA}$. The class $M(A)=\bigcup_{\alpha\geq 1}M_\alpha(A)$ is the ``magmatic universe above $A$.''